Quantum state of fields in SU(∞) quantum gravity

Our Universe is ruled by quantum mechanics and hence should be treated as a quantum system. SU(∞)-QGR is a recently proposed quantum model for the Universe, in which gravity is associated to the SU(∞) symmetry of its Hilbert space. Fragmentation of its infinite dimensional state due to random quantum fluctuations divides the Universe into approximately isolated subsystems. In addition to the parameters of their internal finite rank symmetries, states and dynamics of subsystems are characterized by four continuous parameters and the perceived classical spacetime is their effective representation, reflecting quantum states of subsystems and their relative evolution. At the lowest order, the effective Lagrangian of SU(∞)-QGR has the form of Yang–Mills gauge theories for both SU(∞)—gravity—and internal symmetries defined on the aforementioned 4D parameter space. In the present work, we study more thoroughly some of the fundamental aspects of SU(∞)-QGR. Specifically, we clarify the impact of the degeneracies of the 𝒮𝔘(∞) algebra on the construction of the model, describe mixed states of subsystems and their purification, calculate measures of their entanglement to the rest of the Universe, and discuss their role in the emergence of local gauge symmetries. We also describe the relationship between what is called internal space of SU(∞) Yang–Mills and the 4D parameter space, and analytically demonstrate the irrelevance of the geometry of parameter space for physical observables. Along with these topics, we demonstrate the equivalence of two sets of criteria for the compositeness of a quantum system, and show the uniqueness of the limit of various algebras leading to 𝒮𝔘(∞).